Some applications of the retraction theorem in exterior algebra (Q2536114)

Let \(V\) be \(n\)-dimensional, \(W\) a subspace of \(V\). \(V^*\) and \(W^*\) are the dual spaces. Let \(E(V)\) be the exterior algebra. \(E(W^*)\) is naturally included in \(E(V^*)\). Let \((\,,\,)\) be the natural bilinear map on \(E(V)\times E(V^*)\) and let \(\rfloor\) be the adjoint of left exterior multiplication in \(E(V)\). If \(I\) is an ideal of \(E(V^*)\), set \[ \mathrm{Char}\, I = \{x\in V: x\rfloor I\subset I\};\quad C(I)= (\mathrm{Char}\, I)^\perp. \] \(C(I)\) is a subspace of \(V^*\), the Cartan subspace of \(I\). It is the smallest subspace of \(V^*\) whose exterior algebra contains an ideal which, when embedded in \(E(V^*)\), generates \(I\). The main result is: Let \(I\) be the ideal generated by the linearly independent vectors \(w_1,\ldots,w_n\in V^*\). Let \(\Omega\) be any member of \(\wedge^2 V^*\). Suppose \(p\) is the smallest integer such that \(\Omega^{p+1}\wedge w_1\wedge\cdots \wedge w_s = 0\). Then \(\Omega^p\wedge w_1\wedge\cdots \wedge w_s\) is decomposable and hence of the form \(u_1\wedge\cdots \wedge u_t\) and \(C(I)\) is spanned by \(u_1,\ldots,u_t\).